KBAT Problems
KBAT Problem: Reading a Journey from a Graph
A KBAT problem where a speed–time graph tells a real story, you must interpret gradients and areas, not just read points.
Gradient means acceleration
On a speed–time graph, a sloping line means the object is speeding up or slowing down, and the steeper the line the greater the acceleration. Reading the story, accelerate, cruise, brake, from the shape is the first KBAT step.
Area means distance
The area under a speed–time graph is the distance travelled. Splitting the area into triangles and rectangles and totalling them answers "how far", and comparing areas answers "which phase covered more ground".
Understand the problem
A car's speed–time graph over 30 seconds has three phases: it speeds up uniformly from 0 to 20 m/s in the first 8 s, holds a steady 20 m/s for the next 12 s, then slows uniformly to rest in the final 10 s. Find the acceleration in phase 1, the total distance, and the deceleration in phase 3.
What it really asks: read gradients and areas as a story, not just plot points.
Plan and solve
- Acceleration in phase 1 is the gradient: a = (20 − 0) / 8 = 2.5 m/s².
- Distance is the area under the graph. Phase 1 is a triangle: ½ × 8 × 20 = 80 m.
- Phase 2 is a rectangle: 12 × 20 = 240 m. Phase 3 is a triangle: ½ × 10 × 20 = 100 m.
- Total distance = 80 + 240 + 100 = 420 m. Deceleration in phase 3 is (0 − 20) / 10 = −2 m/s², a magnitude of 2 m/s².
Check and a variant
Check the times add up: 8 + 12 + 10 = 30 s, matching the journey. The average speed is 420 / 30 = 14 m/s, which sits sensibly between 0 and 20 m/s.
A variant the examiner could add: ask for the single constant speed that would cover the same distance in the same 30 s. That is exactly the average speed, 14 m/s, linking the area you found back to a real interpretation.
Frequently asked questions
How do I tell a speed–time graph from a distance–time graph?
Read the vertical axis label first. On a speed–time graph the gradient gives acceleration and the area gives distance.
On a distance–time graph the gradient itself gives speed and area means nothing. Mixing them up is a common lost mark, so name what each axis measures before you calculate anything.
Do I split the area into triangles and rectangles or use a formula?
Splitting into triangles and rectangles is the reliable method and always works, even for irregular graphs. Compute each simple area, then total them.
A trapezium formula can shortcut some shapes, but the split keeps your working visible and easy for the marker to follow, which protects your method marks.
Is deceleration just negative acceleration?
Yes. A downward-sloping section has a negative gradient, so the acceleration is negative; we call its size the deceleration.
In this problem the gradient is −2 m/s², so the deceleration is 2 m/s². Report the magnitude for deceleration but keep the minus sign if the question asks for acceleration directly.